Axially Symmetric Flow with Finite Cavities I

The existence of an axially symmetric flow of an incompressible inviscid fluid moving around an obstacle is proved. The flow is either in a cylindrical pipe or an unbounded region and the cavity may be finite. The proof is based on a variational approach for a non-continuous functional. Essentially is the assumption that the obstacle is starlike with respect to some point on the axis of symmetry. This allows to apply the technique of symmetrization on the stream function.